Set the framework to be a triangulated category with all set indexed coproducts. In "Relative Homological Algebra and Purity in Triangulated Categories", J. of Algebra 227, (2000), pp. 268- 361, (available at the author home page: http://www.math.uoi.gr/~abeligia/) Apostolos Beligiannis said that the homotopy colimit of a diagram of towers whose terms are triangles is also a triangle, provided that the triangulated category has a model; he gives also a reference, namely the Hovey's book: M. Hovey, "Model Categories", Mathematical Surveys and Monographs 63, AMS, Providence, RI, 1999. My problem is that the reference is not precise, and I do not know too much about model categories. Thus my questions are the following: Why is it true that in a triagulated categories with a model, the homotopy colimit of a tower of triangles is a triangle? Where can I find this result in Hovey's book? Finally what about the dual, namely under what extent is it true that the the homotopy limit of a tower of triangles is a triangle?
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2$\begingroup$ First of all, let me remark that, strictly speaking, the statement is ambiguous since there is no homotopy (co)limit functor in triangulated category. The problem is that there's no way to get canonical induced morphisms. You should perhapes make your question precise. Anyway, I understand what you mean, and I bet you won't find such statements explicitly. They can be proved with model categories techniques, but nobody has cared to do it explicitly since once you know the language the proof is trivial. $\endgroup$– Fernando MuroCommented Oct 22, 2012 at 21:15
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3$\begingroup$ There's however a precise theorem which might be enough for you. It asserts in a precise way what you mean and only requires the triangulated category to be the base of a triangulated derivator. It's in one of the appendices of the following paper by Keller-Nicolás arxiv.org/abs/1009.5904 $\endgroup$– Fernando MuroCommented Oct 22, 2012 at 21:19
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